This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A296564 #36 Jan 08 2018 15:17:39
%S A296564 1,2,7,5,2,6,1,8,4,2,0,9,1,1,7,2,1,3,5,9,2,8,4,7,7,2,0,4,7,8,0,1,5,1,
%T A296564 5,1,4,9,3,4,7,6,0,0,3,7,1,0,7,4,9,0,7,5,4,2,7,6,0,2,6,3,7,6,4,9,3,5,
%U A296564 5,3,7,1,6,7,4,1,8,5,8,7,6,2,1,9,0,0,4
%N A296564 Decimal expansion of lim_{k->infinity} (1/k)*Sum_{i=1..k} A293630(i).
%C A296564 From _Jon E. Schoenfield_, Dec 23 2017: (Start)
%C A296564 Starting with the sequence S_0 = {1,2} and extending it one pass at a time as described at A293630 (obtaining S_1 = {1,2,1,1}, S_2 = {1,2,1,1,1,2,1}, etc.), let n_j be the number of terms in S_j; then for j=0,1,2,..., n_j = 2, 4, 7, 13, 37, 73, 145, 289, 865, 1729, 3457, 10369, 20737, 41473, 82945, 248833, 497665, ... (see A291481).
%C A296564 In the algorithm implemented in the PARI program, the variable "build" specifies the number of passes during which the terms of S_j are actually built and stored. The algorithm then uses the terms of S_build to compute the number (n_j) of terms in S_j and their total value (t_j) for each j in build+1..build+n_build. For build=0,1,2,..., the number of decimal digits to which the final ratio t_j/n_j at j = build + n_build matches the actual limit 1.275261842... is 2, 3, 4, 7, 15, 29, 54, 105, 306, 608, 1213, 3629, 7253, 14501, 28995, 86974, 173941, ...
%C A296564 Thus, for example, using build=7, the number of 1s and 2s in the last sequence actually stored, i.e., S_7, is 289, but the number of terms n_j and their total value t_j are computed for every j up through j = build+n_build = 7 + n_7 = 7 + 289 = 296 (both n_296 and t_296 are 104-digit numbers) and the final ratio t_296/n_296 matches the actual limit to 105 decimal digits. (End)
%C A296564 From _Iain Fox_, Dec 23 2017: (Start)
%C A296564 This is the average value of A293630 on the interval n = 1..infinity.
%C A296564 Is this number transcendental? (End)
%H A296564 Iain Fox, <a href="/A296564/b296564.txt">Table of n, a(n) for n = 1..20000</a>
%e A296564 Equals 1.2752618420911721359284772047801515149347600371...
%e A296564 After generating k steps of A293630:
%e A296564 k = 0: [1, 2]; 1.500000000000...
%e A296564 k = 1: [1, 2, 1, 1]; 1.250000000000...
%e A296564 k = 2: [1, 2, 1, 1, 1, 2, 1]; 1.285714285714...
%e A296564 k = 3: [1, 2, 1, 1, 1, 2, ...]; 1.307692307692...
%e A296564 k = 4: [1, 2, 1, 1, 1, 2, ...]; 1.270270270270...
%e A296564 k = 5: [1, 2, 1, 1, 1, 2, ...]; 1.273972602739...
%e A296564 k = 6: [1, 2, 1, 1, 1, 2, ...]; 1.275862068965...
%e A296564 ...
%e A296564 k = infinity: [1, 2, 1, 1, 1, 2, ...]; 1.275261842091...
%o A296564 (PARI) gen(build) = {
%o A296564 my(S = [1, 2], n = 2, t = 3, L, nPrev, E);
%o A296564 print(S);
%o A296564 print(1.0*t/n);
%o A296564 for(j = 1, build, L = S[#S]; n = n*(1+L)-L; t = t*(1+L)-L^2; nPrev = #S; for(r = 1, L, for(i = 1, nPrev-1, S = concat(S, S[i]))); print(S); print(1.0*t/n));
%o A296564 E = S;
%o A296564 for(j = build + 1, build + #E, L = E[#E+1-(j-build)]; n = n*(1+L)-L; t = t*(1+L)-L^2; print(1.0*t/n));
%o A296564 } \\ (gradually increase build to get more precise answers) _Iain Fox_, Dec 23 2017 with help of _Jon E. Schoenfield_
%Y A296564 Cf. A293630.
%K A296564 cons,nonn
%O A296564 1,2
%A A296564 _Iain Fox_, Dec 15 2017